arX

iv:1

205.

4321

v1 [

gr-q

c] 1

9 M

ay 2

012

Gravitationally Generated Interactions

Salvatore Capozziello1,2, Mariafelicia De Laurentis1,2, Luca Fabbri3,4,5, Stefano Vignolo31Dipartimento di Scienze Fisiche, Universita di Napoli “Federico II”,

Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy2 INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy

3DIPTEM Sez. Metodi e Modelli Matematici, Universita di Genova,Piazzale Kennedy, Pad. D, 16129 Genova, Italy

4INFN Sez. di Bologna, Viale B. Pichat 6/2, I-40127 Bologna, Italy5 Dipartimento di Fisica, Universita di Bologna, Via Irnerio 46, I-40126 Bologna, Italy

(Dated: October 8, 2018)

Starting from a 5D-Riemannian manifold, we show that a reduction mechanism to 4D-spacetimesreproduces Extended Theories of Gravity (ETGs) that are direct generalizations of Einstein’s gravity.In this context, the gravitational degrees of freedom can be dealt under the standard of spacetimedeformations. Besides, such deformations can be related to the mass spectra of particles. Theintrinsic non-linearity of ETGs gives an energy-dependent running coupling, while torsion givesrise to interactions among spinors displaying the structure of the weak forces among fermions. Wediscuss how this scheme is compatible with the known observational evidence and suggest thateventual discrepancies could be detected in experiments, as ATLAS and CMS, today running atLHC (CERN). We finally discuss the consequences of the present approach in view of unification ofphysical interactions.

PACS numbers: 04.20.Cv, 04.20.Fy, 04.20.Gz, 04.60.-m

I. INTRODUCTION

So far as we know, there are four fundamental forces in Nature: electromagnetic, weak, strong and gravitationalinteractions. The Standard Model represents well the first three, but not the gravitational interaction, described bythe geometric theory of General Relativity (GR).The Standard Model of Elementary Particles can be considered a successful relativistic quantum field theory both

from particle physics and group-theory point of view; technically, it is a non-Abelian gauge theory (specifically a Yang-Mills theory) associated with the tensor (Cartesian) product of the internal symmetry groups SU(3)×SU(2)×U(1),where the SU(3) color symmetry for quantum chromodynamics, is treated as exact, whereas the SU(2) × U(1)symmetry, responsible for generating the electro-weak gauge fields, is considered to be spontaneously broken. GR isbuilt upon the Einstein equations, a non-linear set of field equations for the spacetime metric. This non-linearity isindeed a source of difficulty for the GR quantization, and with it for most of the conceptual problems that arise fromhaving GR and quantum principles related in a unique framework.Since the Standard Model is a gauge theory where all the fields mediating the interactions are represented by gauge

potentials, the question is why the fields mediating the gravitational interaction (i.e. the gravitational potentials) aredifferent from those of the other fundamental forces. It is reasonable to expect that there may be a gauge theory inwhich the gravitational fields stand on the same footing as the other gauge fields of the Standard Model [1]. As it iswell-known, this expectation has prompted a re-examination of GR from the point of view of gauge theories: whilethe gauge groups involved in the Standard Model are all internal symmetry groups, the gauge groups in GR must beassociated with external spacetime symmetries. Therefore, the gauge theory of gravity cannot be considered underthe same standard of the Yang-Mills theories. It must be one in which gauge objects are not only gauge potentials butalso tetrads that relate the symmetry group to the external spacetime. For this reason, we have to consider a morecomplex non-linear gauge theory where all the interactions should be dealt with under the same standards [2]. In GR,Einstein took the spacetime metric components as the basic set of variables representing gravity, whereas Ashtekarand collaborators employed the tetrad fields and the connection forms as the fundamental variables [3]. We also willconsider the tetrads and the connection forms as the fundamental fields but with the difference that this approachgives rise to a covariant symplectic formalism capable of achieving the result of dealing with physical fields under thesame standard [4, 5].In order to frame historically our approach, let us sketch a quick summary of the various attempts where the

Standard Model and GR have been considered under the same comprehensive picture. In 1956, Utiyama suggestedthat gravitation may be viewed as a gauge theory [6] in analogy to the Yang Mills [7] theory (1954). He identified thegauge potential due to the Lorentz group with the symmetric connection of the Riemann geometry, and reproducedthe Einstein GR as a gauge theory of the Lorentz group SO(3, 1) with the help of tetrad fields introduced in an ad

hoc manner. Although the tetrads were necessary components of the theory (to relate the Lorentz group) adopted

2

as an internal gauge group to the external spacetime, they were not introduced as gauge fields. In 1961, Kibble[8] constructed a gauge theory based on the Poincare group P (3, 1) = T (3, 1) ⋊ SO(3, 1) (the symbol ⋊ representsthe semi-direct product) which resulted in the Einstein-Cartan theory characterized by curvature and torsion. Thetranslation group T (3, 1) is considered responsible for generating the tetrads as gauge fields. Cartan [9] generalizedthe Riemann geometry in order to include torsion in addition to curvature. Sciama [10], and others (Fikelstein [11],Hehl [12, 13]) pointed out that intrinsic spin may be the source of torsion of the underlying spacetime manifold.Since the form and the role of tetrad fields are very different from those of gauge potentials, it has been thought thateven Kibble’s attempt was not satisfactory as a full gauge theory. There have been a number of gauge theories ofgravitation based on a variety of Lie groups [12–18]. It was argued that a gauge theory of gravitation corresponding toGR can be constructed with the translation group alone, in the so-called teleparallel scheme [19]. Inomata et al. [20]proposed that Kibble’s gauge theory could be obtained, in a way closer to the Yang-Mills approach, by consideringthe de Sitter group SO(4, 1), which is reducible to the Poincare group by a group-contraction. Unlike the Poincaregroup, the de Sitter group is homogeneous and the associated gauge fields are all of gauge potential type and by theWigner-Inonu group contraction procedure, one of the five vector potentials reduces to the tetrad. The non-linearapproach to group realizations was originally introduced by Coleman, Wess and Zumino [21, 22] in the context ofinternal symmetry groups (1969). It was later extended to the case of spacetime symmetries by Isham, Salam, andStrathdee [23] considering the non-linear action of GL(4, R), modulus the Lorentz subgroup. In 1974, Borisov, Ivanovand Ogievetsky [24, 25], considered the simultaneous non-linear realization (NLR) of the affine and conformal groups.They stated that GR can be viewed as a consequence of spontaneous breakdown of the affine symmetry, in the sameway that chiral dynamics, in quantum chromodynamics, is a result of spontaneous breakdown of chiral symmetry. Intheir model, gravitons are considered as Goldstone bosons associated with the affine symmetry breaking. As we willsee below, this approach can be pursued in general. In 1978, Chang and Mansouri [26] used the NLR scheme adoptingGL(4,R) as the principal group. In 1980, Stelle and West [27] investigated the NLR induced by the spontaneousbreakdown of SO(3, 2). In 1982 Ivanov and Niederle considered non-linear gauge theories of the Poincare, de Sitter,conformal and special conformal groups [28, 29]. In 1983, Ivanenko and Sardanashvily [30] considered gravity to be aspontaneously broken gauge theory. The tetrads fields arise, in their formulation, as a result of the reduction of thestructure group of the tangent bundle from the general linear to Lorentz group. In 1987, Lord and Goswami [31, 32]developed the NLR in the fiber bundle formalism based on the bundle structure G (G/H ,H) as suggested by Ne’emanand Regge [33]. In this approach, the quotient space G/H is identified with physical spacetime. Most recently, in aseries of papers, Lopez-Pinto, Julve, Tiemblo, Tresguerres and Mielke discussed non-linear gauge theories of gravityon the basis of the Poincare, affine and conformal groups [34–39].As it should be clear from this extensive resume, the idea of an unification theory, capable of describing all the

fundamental interactions of physics under the same standard, has been one of the main issues of modern physics,starting from the early efforts of Einstein, Weyl, Kaluza and Klein [40] until the most recent approaches [41]. Never-theless, the large number of ideas, up to now proposed, which we classify as unified theories, results unsuccessful dueto several reasons: the technical difficulties connected with the lack of a unitary mathematical description of all theinteractions; the huge number of parameters introduced to build the unified theory and the fact that most of themcannot be observed neither at laboratory nor at astrophysical (or cosmological) conditions [42]; the very wide (andseveral times questionable since not-testable) number of extra-dimensions requested by several approaches. Due tothis situation, it seems that unification is a useful (and aesthetic) paradigm, but far to be achieved, if the trend iscontinuing to try to unify interactions by adding more and more ingredients, rarely with an insightful justification. Adifferent approach could be to consider the very essential physical quantities and try to achieve unification withoutany ad hoc new ingredients. This approach can be pursued starting from straightforward considerations which leadto reconsider modern physics under a sort of economic issue: that is, let us try to unifying forces approaching newschemes but without adding new parameters, just exploiting what already is in the underlying set-up to its full extent.So as we said, Utiyama [6] proposed that GR can be seen as a gauge theory based on the local Lorentz group in

the same way that the Yang-Mills gauge theory [7] is developed on the basis of the internal isospin gauge group. Inthis formulation, the Riemannian connection is the gravitational counterpart of the Yang-Mills gauge fields. WhileSU(2), in the Yang-Mills theory, is an internal symmetry group, the Lorentz symmetry represents the local nature ofspacetime rather than internal degrees of freedom. In order to relate local Lorentz symmetry to the external spacetime,we need to solder the local space to the external space. The soldering tools can be the tetrad fields. Utiyama regardedthe tetrads as objects given a priori while they can be dynamically generated [2] and the spacetime has necessarilyto be endowed with torsion in order to accommodate spinor fields [43]. In other words, the gravitational interactionof spinning particles requires the modification of the Riemann spacetime of GR to be a (non-Riemannian) curvedspacetime with torsion. However spinor fields can be viewed as member of Clifford algebra and exist also in flat spaces[44]. Although Sciama used the tetrad formalism for his gauge-like handling of gravitation, his theory fell shortcomingsin treating tetrad fields as gauge fields. Following the Kibble approach [8], it can be demonstrated how gravitation canbe formulated starting from a pure gauge viewpoint. In particular, gravity can be seen as a gauge theory which can

3

be obtained starting from some local invariance, e.g. the local Poincare symmetry, leading to a suitable unificationprescription [2]. However, torsion is not the minimal ingredient we will require, as it is naturally part of the mostgeneral geometry. The minimal ingredient we require is a 5-dimensional extension that, in 4-dimensions, reduces toan enlargement of the Hilbert-Einstein Lagrangian (ETGs): starting from the 5D-space, we will see that the processof reduction to 4D-space generates the mass spectra of particles, and in this picture, we will show that deformationscan be parametrized as effective scalar fields in a GL(4)-group of diffeomorphisms; alternatively, the Hilbert-Einsteinaction R will be taken to be in its most general form as a generic function of the Hilbert-Einstein action f(R). In thiscircumstance, we will see that the factor 1

f ′(R) will give rise to an energy-dependent running-coupling able to bring

effects that are supposed to take place at the Planck scale down to the usual Fermi scale. This is a straightforwardparadigm for ETGs which could involve also other curvature invariants [50, 51]. Here, we are going to eventuallysee that torsion gives rise to spinorial interactions which, in some situations, can be worked out to display the samestructure of the weak forces.The layout of the paper is the following: in Section 2, we shall recall the basic geometrical concepts and clarify the

notations and definitions we will employ in the rest of the paper; in Section 3, we will briefly introduce the 5D-spaces,and their reduction to 4D-spacetimes; Section 4, we discuss deformations of spacetime under the standard of GL(4)-group of transformations. In Section 5, we discuss the reduction from 5D to 4D showing how massive states can comeout. Section 6 is devoted to the generation of masses and their relation to geometrical deformations. In Section 7,we are going to deal with the concept of mass generation and symmetry breaking. The further gravitational degreesof freedom, coming from ETGs, can play a fundamental role in this picture generating natural cut-off at TeV scales:this fact could give rise to gravitationally induced electroweak interactions. In particular, as discussed in Section 8,the weak forces emerge from the interaction of torsion and spinors and a running coupling constant is directly relatedto the form of f(R)-gravity. A detailed discussion of this feature is reported. Conclusions are drawn in Section 9.

II. GEOMETRICAL NOTATIONS AND DEFINITIONS

To begin with, we shall recall some concepts and introduce the notation we will follow. The fundamental objects weare going to work with are tensors, defined in terms of their transformation laws, and whose indices, upper or lower,may be converted into one another, by raising lower or lowering upper indices, by means of the tensors gαβ and gαβrespectively; in order to ensure that the raising and lowering of the indices is consistently defined, these two tensorsare assumed to be non-degenerate, symmetric and one the inverse of the other gνρg

ρµ = δµν , therefore displaying allproperties that make them metric tensors.Dynamical properties of tensors are defined through covariant derivatives Dµ which are themselves defined upon

introduction of the connections Γαµν defined in term of its transformation law alone, and because the most generalconnection is not symmetric in the two lower indices then its antisymmetric part does not vanish and writing it as

T λµν = Γλ[µ,ν] (1)

it turns out to be a tensor called Cartan torsion tensor; in order to ensure that the raising and lowering of the indicesis consistently defined even after differential properties are introduced, these two tensors are assumed vanish theircovariant derivatives Dµg = 0, and therefore we will say that the metric tensors are covariantly constant or verifyingmetricity or again that the metric and connection are compatible: after this condition is taken into account, the mostgeneral connection can be decomposed as

Γµσπ = 12gµρ (∂πgσρ + ∂σgπρ − ∂ρgσπ) +

(T µσπ + T µ

σπ + T µπσ

)(2)

where

Kµσπ = T µσπ + T µ

σπ + T µπσ (3)

is defined as the contorsion tensor, and which may turn out to be useful in some context.Due to their symmetry properties, torsion and contorsion have only one independent component Kµ = Kµπ

π =2T πµ

π = 2T µ and called Cartan torsion or contorsion vectors.The quantity be expressed as

Rρσµν = ∂νΓρσµ − ∂µΓ

ρσν + ΓρλνΓ

λσµ − ΓρλµΓ

λσν (4)

is a tensor antisymmetric in the first and second couple of indices called Riemann curvature tensor; notice that thefirst couple of indices is antisymmetric only after the metricity condition is accounted: this expression of the Riemann

4

curvature tensor has been given so that, together with the expression of the Cartan torsion tensor, we may now writethe commutator of two covariant derivatives as

[Dµ, Dν ]Aρ = T θµνDθA

ρ − 12R

ρσµνA

σ (5)

for the case of a generic vector Aµ, and for which it is understood that for general tensor there will appear anadditional contribution of the Riemann curvature tensor for any additional index. In particular for scalars, theRiemann curvature tensor would not be present at all, thus underlining the specific role of the Cartan torsion tensorin describing the failure of the commutation of the covariant derivatives even for scalar fields.Due to their symmetry properties, curvature has only one independent component that is given by Rρσρν = Rσν

called Ricci curvature tensor, with one component Rσνgσν = R called Ricci curvature scalar.

The commutator of three coordinate covariant derivatives in cyclic permutations gives the expressions

(2DκTρµν + 4T ρκπT

πµν −Rρκµν) + (2DνT

ρκµ + 4T ρνπT

πκµ −Rρνκµ) +

+(2DµTρνκ + 4T ρµπT

πνκ −Rρµνκ) ≡ 0 (6)

known as the Bianchi identities for torsion and

(DµRνικρ − 2RνιβµT

βκρ) + (DκR

νιρµ − 2RνιβκT

βρµ) +

+(DρRνιµκ − 2RνιβρT

βµκ) ≡ 0 (7)

known as Bianchi identities for curvature.Of course, these identities can be contracted, and in their fully contracted form they read as in the following

Dρ

(T ρµν + δρνTµ − δρµTν

)+ 2Tρ

(T ρµν + δρνTµ − δρµTν

)+R[µν] ≡ 0 (8)

known as fully contracted Bianchi identities for torsion and

Dν

(Rνρ − 1

2gνρR

)− 2RνβT

βνρ + TβµνRµνβρ ≡ 0 (9)

known as fully contracted Bianchi identities for curvature.A final remark is to be made for torsion: because torsion is a tensor, then it may be separated away, so that all

most general torsional quantities can be written in terms of simplest torsionless quantities plus torsional corrections.In the following, we will employ the symbols ∇ and R to designate the covariant derivatives and curvatures of thesimplest torsionless connection.Now, given the metric tensors gαβ and gαβ it is possible to define covariant metric concepts like lengths and angles;

let us considered a pair of bases of vectors called beins V aβ and V βa defined to be dual of one another V aµ Vρa = δρµ

and V aµ Vµr = δar which we can always choose to be orthonormal V aσ V

bρ g

σρ = ηab or equivalently V σa Vρb gσρ = ηab

where ηab and ηab are unitary diagonal matrices known as Minkowskian matrices: although it is always possibleto orthonormalize a basis of vectors so that it is without loss of generality that orthonormal beins are introduced,nevertheless the bein are determined up to a Lorentz transformation that can be made explicit. The main advantageof introducing the beins is that, as the metric tensors are used to move indices up or down in tensors, the bein areused to switch to spacetime Greek indices to Lorentz Latin indices, whose explicit transformation laws allow for asimpler description of the formalism; for the moment, this formalism will only give us a simpler way to deal withcalculations, but in the following we will see it will be necessary for the introduction of the spinor fields.According to the definition of bein then, the world tensors are defined in terms of their transformation law under

Lorentz transformations, whose indices, upper or lower, may be converted into one another, by raising lower orlowering upper indices, by means of the Minkowskian matrices, symmetric and inverse of one another ηabη

bm = δmaas expected.Dynamical properties of tensors are defined by covariant derivatives Dµ which are themselves defined upon intro-

duction of the spin-connections Aijµ which is a connection under Lorentz transformation although it is a covector undergeneral coordinate transformations; if we want that the passage between spacetime and Lorentz indices is consistenteven after differential properties are introduced, then we have to assume the condition DαV = 0 analogously to whatwe have done before, and therefore the intuitive condition Dαη = 0 is given automatically: these two conditionstogether imply that the spin-connection is given by

Abjµ = V αj Vbρ

(Γραµ + V kα ∂µV

ρk

)(10)

in terms of the connection and antisymmetric in the two world indices.

5

Considering the spin-connection it is possible to define

Rabσπ = ∂πAabσ − ∂σA

abπ +AajπA

jbσ −AajσA

jbπ (11)

antisymmetric in both spacetime and world indices and such that we have the simple relationship Rabσπ = RµρσπVρb V

aµ

in terms of the Riemann curvature tensor, as it should have been obvious since all definitions have been given so toreduce the passage from the two formalisms a mere renaming of indices; in this formalism, the antisymmetry in thetwo Lorentz indices is manifest. The commutators are given in an equivalent way.As we have anticipated, the introduction of the bein are able to simplify the formalism, but their most important

contribution is the fact that through them we are capable of defining new fields such as the spinors. To see that, itis enough to remark that, whereas nothing can be done for general coordinate transformations, the Lorentz trans-formations have an explicit structure, which can be written in different representations; of all these representations,the most special are the complex-valued representations: complex representations of the Lorentz transformations arecalled spinorial transformations S and they can be expanded in terms of their infinitesimal generators

γij =14 [γi, γj] (12)

where the γa matrices belong to the Clifford algebra.Once the spinorial transformation S is obtained, we define the complex fields that transform according to this

transformation as spinor fields, classified in terms of half-integer spin; for what we will have to do, we will restrict

ourselves to the case given by the fields transforming as ψ′ = Sψ or ψ′= ψS−1, being 1

2 -spin spinor fields, for which

the passage between the two forms is defined by the γ0 matrix as ψ ≡ ψ†γ0 or ψ ≡ γ0ψ†reciprocally.

Dynamical properties for spinor fields are defined through spinorial covariant derivatives, after the introductionof the spinorial connection Ωµ as usual; we notice that for the way in which they have been built, the γj and γijmatrices are constant for the spinorial covariant derivatives automatically: consequently, we have that the mostgeneral spinorial connection can be decomposed as

Ωµ = Aabµγab + iqAµI (13)

in terms of the spin-connection plus an additional term proportional to the identity matrix that will be identified withthe electrodynamic potential, as we shall see in a moment.Given the spinorial connection it is possible to define

Rσπ = ∂σΩπ − ∂πΩσ +ΩσΩπ − ΩπΩσ (14)

antisymmetric and such that Rσπ = Rijσπγij + iqFσπI in terms of the Riemann and Maxwell tensors. Eventually, thecommutators will equivalently be defined as above.It is important to notice that for the decomposition of the most general spinorial connection, the additional term

proportional to the identity matrix should not be surprising: in fact, because spinors are complex fields, then we

have to consider an additional transformation for a complex unitary phase ψ′ = eiqθψ or ψ′= e−iqθψ for a given real

number q called charge, and therefore the issue of their invariance under gauge transformations is to be consideredas well; the theory that is thus constructed is the widely known electrodynamic Maxwell theory. This abelian gaugepotential will eventually be placed in the spinorial connection; and the fact that the spinorial connection already hadenough room to host this abelian gauge field should therefore not be so surprising any longer. In this sense then,electrodynamics has the same mathematical imprint of gravitation, and likewise it should be thought as a genuinegeometrical theory.The reason for which usually it is not can be traced back to a little difference we already underlined, that is the

presence of the charge; however, the charge labels a property of the matter field, and therefore what is not geometricis the individual response of a given matter field propagating in an assigned gauge field, but the electrodynamic fieldin itself might indeed be considered as a purely geometrical field.This concludes the introduction of the formalism, through which we not only defined the fundamental quantities

we will employ in the following, but we also defined them in order to underline the geometrical essence beneath: thisdoes not only refer to the well-known geometrical essence of the gravitational field, but also to the geometrical essenceof the electrodynamic field (at least in terms of the field itself and not the response of a matter field in it), and thegeometric settings of the spinorial fields (before field equations are defined); these ideas are actually not new, and forall concepts recalled here, a general introduction as well as a deeper discussion about the idea of geometrization, canbe found in [45] (additionally it is possible to find references about all different roles of torsion).We notice that so far, the formalism we have introduced is still non-dynamical (we have not defined any field

equation or action) and still valid in any dimension (we have not chosen a particular background); in the following,we will discuss generalized actions for gravity. Specifically, we will take into account a 5D-spacetime with dynamicalreduction to 4D-spacetime.

6

III. THE GEOMETRICAL STRUCTURE OF 5D AND 4D SPACES

In this section, we will discuss the structure of a 5D-Riemannian manifold, and its reduction to 4D-manifold: suchan approach gives a useful tool to deal with the realization of effective theories of gravity in 4D and the problem ofmass generation.Let us start with the 5D-space, where we do not define a priori a signature for the metric and which, after a

4D-reduction procedure, must be capable of reproducing all the features of the usual spacetime [48]. Let us assumethat a 5D-vector field defines a metric whose signature is given by

(X,X) = (x0)2 −−→X · −→X + ǫ(x4)

2 (15)

where ǫ = ±1, so that, using the traditional terms, the fifth dimension can be time-like or space-like: as we shallsee below, it is the 4D-dynamics that discriminates, by a bijective correspondence, the signature, giving rise toparticle-like solutions for ǫ = −1 or to wave-like solutions for ǫ = +1. In this case, we can obtain pseudo-spheres ofLorentz signature and thus 4D-spacetimes of constant curvature (as Friedmann-Robertson-Walker ones). There areonly two independent different signatures for N = 5: they are (p, q) = (1, 4), corresponding to the case ǫ = −1 and(p, q) = (2, 3), corresponding to ǫ = +1. The 5D-manifolds are M(1,4) = R5 and M(2,3) = R5 respectively, whereR5 is the 5D-space, the former case is called the De Sitter space, while the latter is the Anti-De Sitter one. The factthat the standard signature of the universe is (1,−1,−1,−1) can be derived from an equivalent process starting fromM(1,4) or M(2,3). This procedure is in agreement with the Whitney Theorem [49].An important consideration is necessary at this point. After the reduction to 4D-physics, a given theory of gravity

is described by the conformal-affine group of diffeomorphisms GL(4) which is characterized by 4× 4 = 16 generators;a straightforward splitting is

GL(4)︸ ︷︷ ︸4× 4︸ ︷︷ ︸

⊃ SU(3)︸ ︷︷ ︸32 − 1︸ ︷︷ ︸

⊗SU(2)︸ ︷︷ ︸22 − 1︸ ︷︷ ︸

⊗U(1)︸ ︷︷ ︸1︸︷︷︸

⊗GL(2)︸ ︷︷ ︸2× 2︸ ︷︷ ︸

(16)

where the number of generators is indicated for any subgroup: the physical meaning of SU(3), SU(2) and U(1) isclear while GL(2) is a group with 4 generators.The further generators can be recovered in the framework of ETGs being massive or ghost gravitational modes

[48, 50–52]. Clearly the groups of diffeomorphisms GL(4) and GL(2) are not unitary and this could be considereda problem in the standard approaches to quantization. On the contrary, from our point of view, this feature arerelated to mechanisms suitable to induce electroweak interactions and gravitational running coupling constants. Theexperimental consequences of these considerations could be extremely interesting for the physics at LHC.

IV. DEFORMATIONS AND THE PHYSICS OF THE GL(4)-GROUP

Another ingredient for our considerations is represented by the spacetime deformations that can be induced in thereduction mechanism and can be seen as scalar fields in a gravitational background [54]. Let us take into account ametric g on a spacetime manifold M (in N -dimensions); let us now define a new tetrad field ω = ΦAC(x)ω

C , with

ΦAC(x) a matrix of scalar fields, and finally introduce a spacetime M with the metric g defined as

g = ηABΦACΦ

BD ω

CωD = γCD(x)ωCωD, (17)

where also γCD(x) is a matrix of fields which are scalars with respect to the coordinate transformations.

If ΦAC(x) is a Lorentz matrix in any point of M, then g ≡ g, otherwise we say that g is a deformation of g and Mis a deformed M. If all the functions of ΦAC(x) are continuous, then there is a one-to-one correspondence between

the points of M and the points of M. If ξ is a Killing vector for g on M, the corresponding vector ξ on M is not aKilling vector.A particular subset of these deformation matrices is given by

ΦAC(x) = Ω(x) δAC . (18)

which define conformal transformations of the metric g = Ω2(x)g. In this sense, the deformations defined by Eq. (17)can be regarded as a generalization of the conformal transformations.We call the matrices ΦAC(x) First Deformation Matrices, while we can refer to

γCD(x) = ηABΦAC(x)Φ

BD(x). (19)

7

as the Second Deformation Matrices which, as seen above, are also matrices of scalar fields. They generalize theMinkowski matrix ηAB with constant elements in the definition of the metric. A further restriction on the matricesΦAC comes from the above mentioned theorem proved by Riemann for which an N -dimensional metric has N(N−1)/2degrees of freedom (see [53] for details). With this definitions in mind, let us consider the main properties of deformingmatrices. Let us take into account a four dimensional spacetime with Lorentzian signature. A family of matricesΦAC(x) such that

ΦAC(x) ∈ GL(4)∀x, (20)

are defined on such a spacetime. These functions are not necessarily continuous and can connect spacetimes with

different topologies. A singular scalar field introduces a deformed manifold M with a spacetime singularity.As it is well known, the Lorentz matrices ΛAC leave the Minkowski metric invariant and then

g = ηEFΛEAΛ

FBΦ

ACΦ

BD ω

CωD = ηABΦACΦ

BD ω

CωD (21)

and it follows that ΦAC give rise to right cosets of the Lorentz group, i.e. they are the elements of the quotientgroup GL(4,R)/SO(3, 1). On the other hand, a right-multiplication of ΦAC by a Lorentz matrix induces a differentdeformation matrix.The inverse deformed metric is

gαβ = ηCDΦ−1ACΦ

−1BDe

αAe

βB (22)

where Φ−1ACΦ

CB = ΦACΦ

−1CB = δAB and we can decompose the matrix ΦAB = ηAC ΦCB in its symmetric and

antisymmetric parts

ΦAB = Φ(AB) +Φ[AB] = Ω ηAB +ΘAB + φAB (23)

where Ω = ΦAA, ΘAB is the traceless symmetric part and φAB is the antisymmetric part of the first deformationmatrix. Then standard conformal transformations are nothing else but deformations with ΘAB = φAB = 0 [55].

Finding the inverse matrix Φ−1AC in terms of Ω, ΘAB and φAB is not immediate, but as above, it can be split in

the three terms

Φ−1AC = αδAC +ΨAC +ΣAC (24)

where α, ΨAC and ΣAC are respectively the trace, the traceless symmetric part and the antisymmetric part of theinverse deformation matrix above. The second deformation matrix, from the above decomposition, takes the form

γAB = ηCD(Ω δCA +ΘCA + φCA)(Ω δ

DB +ΘDB + φDB) (25)

and then

γAB = Ω2 ηAB + 2ΩΘAB + ηCD ΘCAΘDB +

+ηCD (ΘCA φDB + φCAΘDB) + ηCD φ

CA φ

DB (26)

as it is clear.In general, the deformed metric can be split as

gαβ = Ω2gαβ + γαβ (27)

where

γαβ =(2ΩΘAB + ηCD ΘCAΘDB + ηCD (ΘCA φ

DB+

+φCAΘDB) + ηCD φCA φ

DB

)ωAαω

Bβ (28)

and in particular, if ΘAB = 0, the deformed metric simplifies to

gαβ = Ω2gαβ + ηCD φCA φ

DBω

Aαω

Bβ (29)

so that, for Ω = 1, the deformation of a metric consists in adding to the background metric a tensor γαβ . We have toremember that all these quantities are not independent as, by the theorem mentioned in [53], they have to form at

8

most six independent functions in a four dimensional spacetime. Similarly the controvariant deformed metric can bealways decomposed in the following way

gαβ = α2gαβ + λαβ (30)

identically. To find the relation between γαβ and λαβ we may use gαβ gβγ = δγα, and then

α2Ω2δγα + α2γγα +Ω2λγα + γαβλβγ = δγα (31)

and if the deformations are conformal transformations, we have α = Ω−1, so that assuming such a condition, oneobtains the following matrix equation

α2γγα +Ω2λγα + γαβλβγ = 0 , (32)

and

(δβα +Ω−2γβα)λγβ = −Ω−4γγα (33)

and finally

λγβ = −Ω−4(δ +Ω−2γ)−1αβγ

γα (34)

where (δ +Ω−2γ)−1 is the inverse tensor of (δβα +Ω−2γβα).To each matrix ΦAB we can associate a (1, 1)-tensor φαβ defined by

φαβ = ΦABωBβ e

αA (35)

such that

gαβ = gγδφγαφ

δβ (36)

and viceversa from a (1, 1)-tensor φαβ , we can define a matrix of scalar fields as

φAB = φαβωAα e

βB (37)

as it is to be expected.In conclusion, the spacetime deformations are endowed with the conformal-affine structure of GL(N) groups (in

particular GL(4)) and passing from a given metric to another gives rise to induced scalar fields that, as we shall showbelow, can be generated by a dimensional reduction mechanism.

V. THE 5D-SPACE AND ITS REDUCTION TO 4D-SPACETIME DYNAMICS

We are interested in a reduction mechanism capable of generating the masses of particles and a symmetry breakingcapable of generating the observed interactions. To achieve these goals, we start from a theory of gravity, formulatedin 5D and analyze its reduction in 4D. The results of this approach are ETGs. In the following, 5D indices will becapitalized. We can start from the matter-free case in 5D-gravity. As it turns out, the absence of the Dirac field givesrise to a torsion-free solution: the structure of the field equations thus reduce to that of the usual Einstein gravity.However, it is possible to exploit the fact that the 5D-space gives rise, after the reduction to the 4D-spacetime, toadditional degrees of freedom that can be physically interpreted: these fields arise from deformations as new scalarfields.Now, in 5D-spaces, the field equations are derived from the 5D Hilbert-Einstein action as discussed above; its

variation gives the 5-dimensional field equations in the vacuum

GAB = RAB − 1

2gABR = 0 (38)

where we have defined for simplicity the tensor GAB known as Einstein tensor: here the Ricci-flat space is always asolution.

9

The 5D energy tensor for scalar fields is defined as usual

TAB = ∇AΦ∇BΦ− 1

2gAB∇CΦ∇CΦ (39)

where only the kinetic terms are present; as it is customary, such a tensor can be derived from a variational principlestarting from a Lagrangian LΦ related to the scalar field Φ, whose physical meaning will be clear below.Because of the definition of 5D-space itself, it is important to stress now that no self-interaction potential V (Φ)

has been taken into account so that TAB is a completely symmetric object and Φ is, by definition, a cyclic variable:this fact guarantees that Noether theorem holds for TAB and a conservation law intrinsically exists. With theseconsiderations in mind, the field equations can now assume the form

RAB = χ

(TAB − 1

2gABT

)(40)

where T is the trace of TAB, χ is the 5D gravitational constant and ~ = c = 1. The form (40) of field equations isuseful in order to put in evidence the role of the scalar field Φ, if we are not simply assuming Ricci-flat 5D-spaces.As we said, TAB is a symmetric tensor, and due to the choice of the metric and to the symmetric nature of the

stress-energy tensor TAB and of the Einstein field equations GAB , the contracted Bianchi identities ∇ATAB = 0 hold.

Developing them gives

∇ATAB = ∇A

(∂BΦ∂

AΦ− 12δAB∂CΦ∂

CΦ)=

= (∇AΦB)ΦA +ΦB

(∇AΦ

A)− 1

2 (∇BΦC)ΦC − 1

2ΦC(∇BΦ

C)=

= (∇AΦB)ΦA +ΦB

(∇AΦ

A)− ΦC

(∇BΦ

C)

(41)

and since our 5D-space is a Riemannian manifold, then ∇AΦB = ∇BΦA so that in this case, partial and covariantderivatives coincide for the scalar field Φ. Finally we get

∇ATAB = ΦB

(5)Φ (42)

where (5) is the 5D d’Alembert operator defined as ∇AΦ

A ≡ gAB∇A∇BΦ ≡ (5)Φ.

The general result is that the conservation of the energy tensor TAB implies the Klein-Gordon equation, whichassigns the dynamics of Φ, that is ∇AT

BA = 0 implies that (5)

Φ = 0 assuming of course ΦB 6= 0 since we are dealingwith a non-trivial physical field. As we shall see below, the conservation law of the 5D energy of the scalar field givesa physical meaning to the fifth dimension.The reduction to the 4D-dynamics can be accomplished by taking into account the Campbell theorem [56]: this

theorem states that it is always possible to consider a 4D Riemannian manifold, defined by the line element ds2 =gαβdx

αdxβ , in a 5D one with dS2 = gABdxAdxB . We have gAB = gAB(x

α, x4) with x4 the yet unspecified extracoordinate; as we discussed above, gAB is covariant under the group of 5D coordinate transformations but not underthe restricted group of 4D transformations. This fact has the relevant consequence that the choice of 5D coordinatesresults as the gauge necessary to specify the 4D physics also in its non-standard aspects. Viceversa, in specifying the4D physics, the bijective embedding process in 5D gives physical meaning to the fifth coordinate x4.In other words, the fifth coordinate, in 4D can assume a physical meaning, and we are going to see how this might

be connected to the mass of elementary particles.The reduction to the 4D-spacetime is encoded by the Lagrangian

L =√−g(5)

[(5)R+ λ(g44 − ǫΦ2)

](43)

with λ as a Lagrange multiplier, Φ a scalar field and ǫ = ±1. This approach is completely general and used intheoretical physics when we want to put in evidence some specific feature [57]; in this case, we need it to derive thephysical gauge for the 5D-metric.We can write down the metric as

dS2 = gABdxAdxB = gαβdx

αdxβ + g44(dx4)2 = gαβdx

αdxβ + ǫΦ2(dx4)2 , (44)

from which we obtain directly particle-like solutions for ǫ = −1 or wave-like solutions for ǫ = +1 in the 4D-reductionprocedure. The standard signature of 4D-component of the metric is (+1,−1,−1,−1) and α, β = 0, 1, 2, 3.Furthermore, the 5D-metric can be written in a Kaluza-Klein fashion as the matrix

gAB =

(gαβ 00 ǫΦ2

), (45)

10

and the 5D-curvature Ricci tensor can be derived. After the projection from 5D to 4D, gαβ , derived from gAB, nolonger explicitly depends on x4, and a useful expression for the Ricci scalar can be derived:

(5)R = R− 1

ΦΦ , (46)

where the dependence on ǫ is explicitly disappeared and is the 4D-d’Alembert operator, which gives Φ ≡gµν∇µ∇νΦ.The action above can therefore be written in a 4D-reduced Brans-Dicke action of the following form

L =1

16πGN

√−g [ΦR+ LΦ] (47)

where the Newton constantGN is given in terms of the 5D gravitational constant and a characteristic length associated

to the Compton length as GN =(5)G2πl . Defining the function of the 4D-scalar field φ as

− Φ

16πGN= φ (48)

we get in 4D a general action in which gravity is non-minimally coupled to a scalar field as

A =∫M d4x

√−g[φR + 1

2gµν∇µφ∇νφ− V (φ)

]+∫∂M d3x

√−bK (49)

where the form and the role of V (φ) are still general. The second integral is a boundary term where K ≡ hijKij isthe trace of the extrinsic curvature tensor Kij of the hypersurface ∂M embedded in the 4D-manifold M and b is themetric determinant of the 3D-manifold.The Einstein field equations are derived by varying with respect to the 4D-metric gµν

Rµν − 12gµνR = Tµν (50)

where

Tµν = 1φ

[− 1

2∇µφ∇νφ+ 14gµν∇αφ∇αφ− 1

2gµνV (φ)− gµνφ+∇µ∇νφ]

(51)

is the effective energy tensor containing the non-minimal coupling contributions, the kinetic terms and the potentialof the scalar field φ.By varying with respect to φ, we get the 4D-Klein-Gordon equation, which are nothing else but the contracted

Bianchi identity, as we have already shown; this implies that the effective energy tensor in the right-hand side of (50)is a divergenceless tensor, which is compatible with Einstein theory of gravity also if we started from a 5D-space, aslong as the theory is torsionless, that is in the vacuum of spinor fields.Finally in order to physically identify the fifth dimension, we may write the Klein-Gordon equation in the form

(+m2

eff

)φ = 0 (52)

where

m2eff = 1

φ [V ′(φ)−R] (53)

is the effective mass given in terms of scalar field self-interactions and external gravitational contributions. In anyquantum field theory formulated on curved spacetimes, these contributions, at one-loop level, have the same weight,and indeed, Extended Theories of Gravity are renormalizable at one loop-level [58].What we want to show next is that a natural way to generate particle masses can be achieved starting from a 5D

picture. In other words, the concept of mass can be derived from a geometric viewpoint.

VI. THE GENERATION OF MASSES

A. Particle masses as eigenstates from 5D-spaces

The above considerations allow a straightforward mechanism for the generation of masses that can be related tothe projection from 5D to 4D-manifolds.

11

In a 5D-space, the 5D d’Alembert operator for particle-like solutions can be split selecting the value ǫ = −1 in themetric as (5)

= − ∂42, so that for a given scalar field Φ, we have

(5)Φ =

[− ∂4

2]Φ = 0 (54)

identically; because the (45 ) is diagonal in the fifth component, we may separate the variable as Φ = φ(t, ~x)ψ(x4)and thus

φφ = 1

ψ

[d2ψdx2

4

]= −k2n (55)

where kn must be a constant. From Eq.(55), we obtain the two equations of motion(+ k2n

)φ = 0 (56)

d2ψdx2

4+ k2nψ = 0 . (57)

the former being the evolution equation for the scalar field with a mass term.The constant kn has the physical dimension of the inverse of a length and, assigning boundary conditions, we can

derive the eigenvalue relation kn = 2πl n where n is an integer. As a result, in standard units, we recover the Compton

length λn = ~

2πmnc= 1

knwhich assigns the mass of a particle. It has to be stressed that, the eigenvalues of Eq.(57)

are the masses of particles which are generated by the reduction process from 5D to 4D. On the other hand, differentvalues of n fix the families of particles, while, for any given value of n.

B. 4D-spacetimes and dynamics of massive scalar fields

Dynamics of 4D-component of the induced scalar field Φ can be related to the spacetime deformations discussedabove; in this way, the role of GL(4)-group of diffeomorphism will result of fundamental importance. Let us startby showing how particles can acquire mass by deformations and let us relate the procedure with the above reductionmechanism.As usual, a particle with zero mass is characterized by the invariant relation ηαβpαpβ = 0 in the Minkowski

spacetime. Deforming the spacetime and considering the above operators, one has

ηABΦCAΦDBe

αCe

βDpαpβ = gαβpαpβ = 0 , (58)

so we have defined two frames, one, the Minkowski spacetime, defined by the metric ηαβ and the other defined by themetric gαβ , generated by the projection from 5D. The two frames are related by the matrices of deformation functionsΦCA(x). In both frames the massless particle follow a null path, but we observe that using the decomposition (30) theparticle does not appear massless with respect to the first frame. As matter of fact, Eq. (58) becomes

Ω−2ηαβpαpβ + χαβpαpβ = 0 , (59)

so that

ηαβpαpβ = −Ω2χαβpαpβ 6= 0. (60)

and now in the first frame we are able to define a rest reference system for the particle.For a massless particle it is not possible to define a rest reference system since considering the invariant relation,

g00p20 + 2g0ip0pi + gijpipj = 0 , (61)

and defining as rest frame the system in which pj = 0, the solution exists only for p0 = 0 i.e. only the trivial solutionpα = 0 satisfies the “ rest reference frame” condition.On the other hand, if we consider massive particle, then

g00p20 + 2g0ip0pi + gijpipj = m2 , (62)

the rest frame is characterized by the conditions pj = 0 and consequently g00p20 = m2. This means that the (squared)mass is proportional to the (squared) energy and the solution is no more trivial. To overcome this problem, let us fixthe deformation such that

−Ω2χαβpαpβ = m2 , (63)

12

m being the mass ”attribute” to the particle. In the second frame we have

p20 − ~p 2 +Ω2χ00p20 + 2χ0ip0pi + χijpipj = 0 , (64)

if ~p = 0 then p20(1 + Ω2χ00) = 1 which implies, besides the trivial solution, also the condition Ω2χ00 = −1 which,together with Eq. (63), gives p20 = m2, the rest system condition in the first frame.It is also possible to show that there is an equivalence between deforming a metric and giving mass to a massless

particle. As we have shown, we have described the spacetime deformation by using matrices of scalar fields. Thesame problem can be addressed in terms of spacetime tensors

ηABΦACΦ

BDe

Cµ e

Dν = gαβΦ

αµΦ

βν = gµν . (65)

It should be observed that the Φαµ do not represent coordinates transformations as far as they cannot be reduced toJacobian matrices. This means that Φαµ have a fundamental physical meaning. Starting from g we observe that ifgµν p

µ pν = 0 then gµν pµ pν 6= 0 when g is a general deformation that can be related to the conformal transformations

of g. Eq. (65) tells us that, equivalently, we can read the deformation as a transformation of the 4-momentum of theparticle

gµν pµ pν = gαβΦ

αµΦ

βνpµ pν ≡ gαβ p

α pβ , (66)

in such a way there exists deformations of spacetime which give mass to massless particles. In other words, deforma-tions, that are elements of the conformally-invariant GL(4)-group can be related, in principle, to the generation ofthe masses of particles.

C. Mass Generation from Deformations

So far we have considered the geometrical definition of mass from the point of view of relativistic mechanics. Nowwe would like to extend this result to classical field theory with the aim to extend it to quantum field theory.Let us consider a scalar free massless particle. It can be described by the Lagrangian

L =1

2

√−ggµν(∂µφ)(∂νφ) =1

2

√−g (ηµν + χµν) (∂µφ)(∂νφ). (67)

It is well-known that the free propagator has a pole in p2 = 0. In order to introduce a mass term in the Lagrangianand in the field equations, we have to eliminate a divergence from it, that is considering

∂µ[∂ν

√−g χµνφ2]= ∂µ∂ν

(√−g χµν)φ2 +

+4 ∂ν(√−g χµν

)φ∂µ φ+ 2

√−g χµν∂µφ ∂νφ+ 2√−g χµνφ ∂µ∂νφ , (68)

the Lagrangian takes the form

L =1

2

√−g (ηµν) (∂µφ)(∂νφ)−1

4∂µ∂ν

(√−g χµν)φ2 −

+ ∂ν(√−g χµν

)φ (∂µ φ)−

1

2

√−g χµνφ ∂µ∂νφ.(69)

In this way a “mass” termm2 = 12∂µ∂ν (

√−g χµν) can be defined in a new Lagrangian. On the other hand, considering

an action A =∫ √−gL and a variational principle δA

δφ = 0 implies the equation

∂L∂φ

− ∂α∂L

∂(∂αφ)+ ∂α∂β

∂L∂(∂α∂βφ)

= 0 , (70)

which gives

φ = ηµν∂µ∂νφ+Ω2χµν∂µ∂νφ = 0 . (71)

It is well-known that this equation cannot give, in general, a potential or a mass term, except when we take as asolution a plane wave φ = exp ikµx

µ. In this case the χ part of the equation can be interpreted as a mass termaccording to

χµν∂µ∂νφ = −Ω2χµνkµkν exp ikµxµ = m2φ . (72)

13

This equation can be compared with Eq. (63), previously derived for a relativistic particle. We can extend this resultto each function which can be expressed by a Fourier transform,

φ(x) =

∫exp(ikµx

µ)f(k)d4k , (73)

where the spacetime dependent mass term is defined by the equation

m2(x) = −∫χµνkµkν exp(ikµx

µ)f(k)d4k. (74)

With these restrictions, the Klein-Gordon equation for a massless particle in the g-frame is seen in the η-frame as amassive particle as soon as the plane wave solutions are considered. This result makes more sense when the mass isinterpreted as a quantum effect. In the case a curved spacetime, the above arguments can be implemented by thesubstitutions of operators η → g and ∂ → ∇. It is important to note that the scalar field equation is not conformallyinvariant. In order to have a conformally invariant scalar field equation, it is necessary to introduce, in the Lagrangiandensity a non-minimal coupling between geometry and field. A possible choice is

L =1

2

√−ggµν(∂µφ)(∂νφ)− ξRφ2 , (75)

By varying with respect to φ, the Klein-Gordon equation

φ+ ξRφ = 0 , (76)

is recovered. Minimal coupling is obtained for ξ = 0. By analogy with the previous considerations, a deformationwith constant curvature R = m2 gives a mass term in the original frame. However, we need also to interpret theother terms appearing in the new frame. Expanding Eq. (76),

ηµν∂µ∂νφ+ χµν∂µ∂νφ+ (ηµν + χµν) Γλµν∂λφ+ ξRφ = 0 , (77)

we need that the connection is compatible with the metric tensor (ηµν + χµν), that is

∇ (ηµν + χµν) = 0. (78)

In order to have a deformation defining a mass m2 = R in the original frame, two conditions must be satisfied,R = R0 > 0 where R0 is a constant and

χµν∂µ∂νφ+ (ηµν + χµν) Γλµν∂λφ = 0 , (79)

which is a restriction on the deformation tensor χµν . This condition allows to determine the mass by a conformaltransformation. In fact, in the new frame, the equation is

gαβ∇α∂βφ+ ξRφ = 0 , (80)

which, in the case of a conformal transformation gαβ = Ω2ηαβ becomes

Ω−2ηαβ(∂α∂βφ+ 2Ω−1∂γΩ∂γφ

)− 6ξηαβΩ−3∂α∂βΩ = 0 . (81)

It is equivalent to a massive scalar field equation (in the first frame), if the condition

2Ω−1∂γΩ∂γφ− 6ξηαβΩ−1∂α∂βΩ = m2 , (82)

holds. Since Eq. (81) is linear in φ, we can define φ = eiλαxα

. Eq. (82) becomes

2Ω−1∂γΩλγ − 6ξηαβΩ−1∂α∂βΩ = m2 , (83)

then a solution is Ω = eikαxα

and the mass depends on the combination

m2 = 6ξηαβkαkβ − 2ηαβkαλβ . (84)

In this example, the deformation is a specific complex function, but we can extend this result to any function bytaking

Ω(x) =

∫Ω(k)eikxd4k , (85)

14

and also non-constant masses can be is obtained being

m2 =

∫Ω(k)(6ξηαβkαkβ − 2ηαβkαλβ)e

ikxd4k∫Ω(k)eikxd4k

. (86)

These results mean that a geometrical definition of mass is always possible. It can be induced by the conformaltransformation Ω a restriction of the deformations group GL(4).Summarizing we have shown that the fifth dimension of a reduction mechanism from 5D to 4D-manifolds can

be interpreted as a mass generator, where the degrees of freedom coming from ETGs have a physical meaning andcannot be simply gauged away, and as soon as particles acquire masses GL(4) can induce symmetry breaking. Inother words, the further degrees of freedom related to the ETGs may be related to a Higgs-like mechanism as we aregoing to discuss below.

VII. GRAVITATIONAL MASSIVE STATES AND INDUCED SYMMETRY BREAKING

The above results could be interesting to investigate quantum gravity effects and symmetry breaking in the rangebetween GeV and TeV scales. Such scales are actually investigated by the experiments at LHC. It is importantto stress that any ultra-violet model of gravity (e.g. at TeV scales) have to explain also the observed weakness ofgravitational effects at largest (infra-red) scales. This means that massless modes have to be considered in any case.The above 5D-action is an example of higher dimensional action where the effective gravitational energy scale

(Planck scale) can be rescaled according to Eq. (48). In terms of mass, being M2p = c~

GN

the constraint coming from

the ultra-violet limit of the theory (1019 GeV) , we can set M2p = MD−2

cut−offVD−4, where VD−4 is the volume comingfrom the extra dimension. It is easy to see that VD−4, in the 5D case, is related to the fifth component of Φ. Mcut−offis the cut-off mass that becomes relevant as soon as the Lorentz invariance is violated. Such a scale, in the contextdiscussed here, could be of the order TeV [61].As we have shown above, it is quite natural to obtain effective theories containing scalar fields of gravitational

origin. In this sense, Mcut−off is the result of dimensional reduction. To be more explicit, the 4D dynamics is ledby the effective potential V (φ) and the non-minimal coupling F (φ). Such functions could be tested since they arerelated to massive states.In particular, the effective Extended Gravity, produced by the reduction mechanism from 5D to 4D, can be chosen

as

A =

∫d4x

√−g[− φ2

2R +

1

2gµν∂µφ∂νφ− V

](87)

plus contributions of ordinary matter terms. The potential for φ can be assumed as

V (φ) =M2cut−off2

φ2 +λ

4φ4 , (88)

where a massive term and the self-interaction term are present. This is the standard choice of quantum field theorywhich perfectly fits with the arguments of dimensional reduction. Let us recall again that the scalar field φ is notput by hand into dynamics but it is given by the extra degrees of freedom of gravitational field generated by thereduction process in 4D. It is easy to derive the vacuum expectation value of φ, being M2

cut−off = 2λM2p , which is a

fundamental scale of the theory.Some considerations are in order at this point. Such a scale has to be confronted with Higgs vacuum expectation

value which is 246 GeV and then with the hierarchy problem. If Mcut−off is larger than Higgs mass, the problem isobviously circumvented. It is important to stress that hierarchy problem occurs when couplings and masses of effectivetheories are very different than the parameters measured by experiments. This happen since measured parametersare related to the fundamental parameters by renormalization. However cancellations between fundamental quantitiesand quantum corrections are necessary, then the hierarchy problem is a fine-tuning problem.In particle physics, the puzzle is why the weak force is stronger and stronger than gravity. Both of these forces

involve fundamental constants of nature, the Fermi constant for the weak force and the Newton constant for gravity.From the Standard Model of Particles, the Fermi constant is unnaturally large and should be of the order of theNewton constant, unless there is a cancellation between the bare value of Fermi constant and the quantum correctionsto it.More technically, the question is why the Higgs boson is lighter and lighter than the Planck mass. In fact, people

are searching for Higgs masses ranging from 115 up to 350 GeV with different selected decay channels from bb to tt

15

(see for example [59] and references therein). One would expect that the large quantum contributions to the square ofthe Higgs boson mass would inevitably make the mass huge, comparable to the scale at which new physics appears,unless there is a fine-tuning cancellation between the quadratic radiative corrections and the bare mass. With thisstate of art, the problem cannot be formulated in the context of the Standard Model where the Higgs mass cannotbe calculated. In a sense, the problem is solvable if, in a given effective theory of particles, where the Higgs bosonmass is calculable, there are no fine-tunings. If one accepts the big-desert assumption and the existence of a hierarchyproblem, some new mechanism (at Higgs scale) becomes necessary to avoid fine-tunings.The model which we are discussing contains a ”running” scale that could avoid to set the Higgs scale with great

accuracy. If the mass of the field φ is in TeV region, there is no hierarchy problem being φ a gravitational scale. Inthis case, the Standard Model holds up plus an extended gravitational sector that could be derived from the fifthdimension. In other words, the Planck scale can be dynamically derived from the vacuum expectation value of φ. Inour model, the Planck scale can be recovered, as soon as the coupling λ is of order 10−30÷31.Considering again the problem of mass generation, one can assume that particles of Standard Model have sizes

related to the cut off, that is M−1cut−off , and their collisions could lead to the formation of bound states as in [60, 66].

Potentially, such a phenomenon could mimic the decay of semi-classical quantum black holes and, at lower energies,it could be useful to investigate substructures. This means that we should expect some strong scattering effects inthe TeV region involving the coupling of φ to the Standard Model fields. The ”signature” of this phenomenon couldlead to polarization effects of the particle beam as discussed in [48]. Furthermore the strong dynamics derived fromthe phenomenon could resemble compositeness as discussed in [68]. These considerations are extremely important inview of Higgs sector physics.In fact the Higgs mechanism is an approach that allows to generate the masses of electroweak gauge bosons; to

preserve the perturbative unitarity of the S-matrix; to preserve the renormalizability of the theory. The masses ofthe electroweak bosons can be written in a gauge invariant form using either the non-linear sigma model [22] or agauge invariant formulation of the electroweak bosons. However if there is no propagating Higgs boson, quantumfield amplitudes describing modes of the electroweak bosons grow too fast violating the unitarity around TeV scales[74–77]. There are several ways in which unitarity could be restored but the Standard Model without a Higgs bosonresults non-renormalizable.A possibility is that the weak interactions become strongly coupled at TeV scales and then the related gauge

theory becomes unitary at non-perturbative level. Another possibility for models without a Higgs boson consists inintroducing weakly coupled new particles to delay the unitarity problem into the multi TeV regime where the UVlimit of the Standard Model is expected to become relevant. In [78], it is proposed that, as black holes in gravitationalscattering, classical objects could form in the scattering of W-bosons.This idea shows several features of electroweak interactions. First of all, the Higgs mechanism is strictly necessary

to generate masses for the electroweak bosons. Beside, some mechanisms can be unitary but not renormalizable orvice-versa. In summary, the paradigm is that three different criteria should be fulfilled: i) a gauge invariant generationof masses of electroweak bosons, ii) perturbative unitarity; iii) renormalizability. The approach we are proposinghere is based on Extended Theories of Gravity deduced from a 5D-manifold reduction where the Standard Model isfully recovered enlarging the gravitational sector and avoiding the hierarchy problem.It is important to point out that, in both the non-linear sigma model and in gauge invariant formulation of Standard

Model, it is possible to define an action in terms of an expansion in the scale of the electroweak interactions v. Theaction can be written as

A = ASMw/oHiggs +

∫d4x

∑

i

CivN

O4+Ni , (89)

where O4+Ni are operators compatible with the symmetries of the model. The analogy between the effective action

for the electroweak interactions (89) and that of Extended Gravity is striking. Considering only the leading terms,the above theory can be written as a Taylor series of the form

f(R) ≃ Λ +∑

k

1

k!

dkf(R)

dRk

∣∣∣∣0

Rk (90)

where the coefficients are the derivatives of f(R) calculated at a certain value of R. Clearly, as shown in previoussections,, the extra gravitational degrees of freedom can be suitably transformed in a scalar field φ which allows toavoid the hierarchy problem. Both electroweak theory and Extended Gravity have a dimensional energy scale whichdefines the strength of the interactions. The Planck mass sets the strength of gravitational interactions while the weakscale λ determines the range and the strength of the electroweak interactions. As shown in the previous subsection,these scales can be compared at TeV energies.

16

In other words, the electroweak bosons are not gauge bosons in standard sense but they can be ”derived” fromthe above further gravitational degrees of freedom. The local SU(2)L gauge symmetry is imposed at the level ofthe quantum fields. However there is a residual global SU(2) symmetry, i.e. the custodial symmetry. In the case ofgravitational theories formulated as the GL(4)-group of diffeomorphisms, tetrads are an unavoidable feature necessaryto construct the theory. They are gauge fields which transform under the local Lorentz transformations SO(3, 1)and under general coordinate transformations, the metric gµν = eaµe

bνηab which is the field that is being quantized,

transforms under general coordinate transformations which is the equivalent of the global SU(2) symmetry for theweak interactions (in our case the residual GL(2) ⊃ SU(2)) . Such an analogy between the tetrad fields and the Higgsfield is extremely relevant. As shown above for deformations, we can say that the Higgs field has the same role ofthe tetrads for the electroweak interactions while the electroweak bosons have the same role of the metric. Dynamicswould be given by deformations.A gravitational action like (90) is, in principle, non-perturbatively renormalizable if, as shown by Weinberg, there

is a non-trivial fixed point which makes the gravity asymptotically free [82]. This scenario implies that only a finitenumber of the Wilson coefficients in the effective action would need to be measured and the theory would thus bepredictive and probed at LHC.In conclusion, the unitarity problem of the weak interactions could be fixed by a non-trivial fixed point in the

renormalization group of the weak scale. A similar mechanism could also fix the unitarity problem for fermionsmasses [87–91] if their masses are not generated by the standard Higgs mechanism but in the same way consideredhere (let us remind that also SU(3) could be generated by the splitting of GL(4) group). In the case of electroweakinteractions this approach could be soon checked at LHC but good indications are also available for QCD [92].

VIII. f(R)-GRAVITY WITH TORSION AND SPINOR FIELDS

Considering torsion in the discussion means to take into account also spin fields [13]. In particular, matter fieldscan be given by Dirac fields: Dirac fermions are defined in terms of gamma matrices satisfying the Clifford algebra.Such fields can be achieved in the process of 4D-reduction considering the whole geometric budget where genericconnections are assumed (not only symmetric ones). The fifth gamma matrix is a parity-odd gamma matrix whoseindex has no geometrical meaning: it is γ5 = iγ0γ1γ2γ3 in terms of the other independent gamma matrices. Thedynamics of the gravitational field can be taken in a more general instance: that is instead of considering the usualHilbert-Einstein Lagrangian R, we will take a general function of the Hilbert-Einstein Lagrangian f(R) derived fromthe reduction process [48]. The special case given by the Einstein-Hilbert Lagrangian is immediately recovered forf(R) = R, but in this case the peculiar form of the Lagrangian is such that yields an Hessian determinant null, andtherefore it is degenerated with respect to all other more general f(R)-theories. In the following we will focus onthese most general f(R)-theories of gravitation which can be always conformally related to the presence of scalarfields [50, 51]. Dirac fermions possess in general both energy and spin density, and so their coupling to both metricand torsion is very natural: for them, it is possible to write an effective Lagrangian as

L =√−g

[f(R) + i

2

(ψγαDαψ − ψγαDαψ

)−mψψ

](91)

that is the Dirac action plus a gravitational action given by a general function f(R) of the Hilbert-Einstein action ofgravity. Note that now we are including torsion degrees of freedom. By varying this action with respect to the fieldsinvolved, we get the field equations in the form

iγαDαψ + iγαTαψ −mψ = 0 (92)

where m is the mass of the Dirac matter field and Tα is the contorsion vector (see Section 2). Energy and spin densityare

Tαβ = i4

(ψγαDβψ −Dβψγαψ

)(93a)

Sµαβ = i4 ψ γµ, [γα, γβ]ψ (93b)

the spin density tensor being completely antisymmetric; the field equations for gravity are

f ′(R)Rαβ − 12f(R)gαβ = Tαβ (94a)

f ′(R)Tµαβ − 14 (gαµ∂βf

′(R)− gβµ∂αf′(R)) = 1

2

(Sαβµ + 1

2gαµSρ

βρ − 12gβµS

ραρ

)(94b)

After substitution of energy and spin density into these field equations, we get

f ′(R)Rαβ − 12f(R)gαβ = i

4

(ψγαDβψ −Dβψγαψ

)(95a)

f ′(R)Tµαβ − 14 (gαµ∂βf

′(R)− gβµ∂αf′(R)) = i

8 ψ γµ, [γα, γβ ]ψ (95b)

17

linking the Dirac matter to the spacetime geometry. Here f ′(R) means the ordinary derivative with respect to R. Wesee that when the Dirac matter field equations are used for the energy and spin density, the geometric field equationsreduce to the Bianchi identities, showing that the model is consistently defined. It is important to stress that we areagain in the realm of ETGs and that f(R)-gravity is nothing else but a scalar-tensor gravity where the 4D-scalar fieldφ has been suitably recast in terms of the curvature scalar R [specifically φ→ f ′(R)] [67].It is possible to see that from (94b) there could be torsion even in absence of the spin density tensor: therefore we

may say that while the energy is the source of the curvature of the spacetime, both the spin and the non-linearity off(R) gravity are the sources of the spacetime torsion, in particular the non-linearity being the source of the torsiontrace vector; however in the case of Dirac field, because of the complete antisymmetry of the spin, matter is the sourceof the completely antisymmetric part of torsion alone.In the following, we suppose that the trace of the field energy-curvature coupling equation (94a) given by f ′(R)R−

2f(R) = Tαα = T is an invertible relation between the Ricci curvature scalar R and the trace of the stress-energytensor; since f(R) = kR2 is only compatible with the traceless energy condition, then we assume that f(R) 6= kR2.Clearly, it is now possible to split all torsional quantities into torsionless quantities plus torsional contributions.

The Dirac theory is algebraically related to the spin, therefore we can substitute torsion with the spin everywhere, toget the Dirac field equations

iγα∇αψ − 316ϕ ψγ

αψγαψ −mψ = 0 (96)

where ∇ designates the purely metric derivatives and the contributions of torsion are written in terms of the spin ofthe spinor field itself. Here have introduced the field ϕ = f ′(R): this means that Dirac field equations1 , in the mostgeneral case with torsion, are dynamically equivalent to those with no torsion but with additional self-interactionpotentials (see consideration in the previous sections). By employing the Fierz rearrangements they can be writtenin the form of Nambu–Jona–Lasinio potentials; moreover this means that Dirac field equations in the most generalf(R)-gravity are dynamically equivalent to those in the simplest case f(R) = R in which the additional potentialsof self-interaction are scaled by a factor ϕ, which may be either positive or negative yielding repulsive or attractiveNambu-Jona–Lasinio potentials respectively [46, 47].In the 4D case, and in absence of spinor fields, it is possible to work out the field equations that we would obtain

by varying with respect to the metric only, as in the purely metric case, that is

f ′(R)Rαβ − 1

2f(R)gαβ = (gαµgβν − gαβgµν)∇µ∇νf ′(R) , (97)

which are fourth-order equations; after a suitable manipulation, the above equation can be rewritten as

Rαβ − 1

2Rgαβ =

1

φ

[1

2gαβ (f(R)−Rf ′(R)) +∇α∇βf

′(R) + gαβf′(R)

](98)

where the right-hand side can be interpreted as an effective energy tensor constructed by the extra curvature terms[50, 51].

A. Torsion and weak interactions

Having established a general framework in which the gravitational background can be related to the electroweakphenomenology, we shall now focus on a specific example in which such an approach can be realized. It is easy to seethat the field 1

ϕ plays the role of an energy-dependent scaling. Let us assume that the action (90) is truncated at the

fourth-order power: the Taylor coefficients can be written as

f(R) = R + 14εR

2 + 127η

2ε2R3 (99)

in terms of the two parameters ε and η that have to be determined. The Λ term has been discarded since it isunessential for the following considerations. In this situation, the trace of the energy-curvature coupling is a cubicequation in R with three solutions, of which we are going to take into account the only real one; this is also the only

1 Here we are using the symbol ϕ instead of φ since torsional degrees of freedom are present in f(R).

18

solution for which R vanishes as the trace of the energy-momentum tensor Σ vanishes. In this sense, this is the correctinfrared limit. Explicitly ϕ is

ϕ = 3(1− 3

8η2

)+(

34η +

3√s+ 2

√s2 − 1

)2

+(

34η +

3√s− 2

√s2 − 1

)2

(100)

function of the energy-momentum trace 12ηεΣ = s in the only parameter η. This 1

ϕ energy-dependent coupling starts

from the unity in the infrared regime, and then it increases up to its maximum value reached at some scale thatmay be fine-tuned eventually, before decreasing to vanish asymptotically in the ultraviolet regime. Henceforth anyNambu-Jona–Lasinio interaction will be negligible in the infrared, then it would get larger up to a given energy, beforeensuring asymptotic freedom in the ultraviolet regime.Having established how the non-linearity of f(R)-gravity behaves as an energy-dependent running coupling for a

given Nambu-Jona–Lasinio interaction, we may now leave the treatment of the single-field in self-interaction to studythe couple of fields in interaction. In this case, the role of torsion is drastically important for the following reason:each spinor has a dynamics governed by a covariant derivative that contains torsion related to the spin density of thetotal system. If we now have only two spinor fields in a region of spacetime getting close to one another, it is clearthat either of them will have dynamical behavior influenced by the presence of the other one, according to whetherthe other spinor is actually close enough to make relevant contributions to the spin density or far enough to considerthose contributions to the spin density negligible. This means that the presence of torsion for a system of two ormany spinors has effects for their dynamics, which make them look like interacting even if no other interaction hasbeen defined.The presence of the running coupling may fine-tune the scale of the interaction to any value. Consequently it is

straightforward to wonder if this torsionally induces interaction might be similar to the nuclear interactions, somehow.These torsional potentials are similar to the forces in the Fermi model: this hypothesis was conjectured since the 1970sin some seminal works [12, 13, 45, 93–95] although one of the problems people had to face was precisely that of theenergy-dependent scale: since this problem appears to be addressable by simply setting the non-linearity of the presentgravitational action, we may ask ourselves what are the possible effects of torsional interactions and, in particular, ifthey may look like the Fermi nuclear interactions.

B. The massless spinor-semispinor system

Let us consider now massless fields, so to split the left-handed and right-handed projections. As a first example, weshall deal with the case in which one of the two spinors is a semispinor having only the left-handed projection [96].The aim is to compare this case with that of the electroweak interactions for leptons before the symmetry breaking.In the case of massless spinors of which one is a spinor and the other is a semispinor (having only the left-handedprojection) we set ψ1

R ≡ 0. The field equations are

iγµ∇µψ1L + 3

16

(ψ1Lγµψ

1Lγ

µψ1L + ψ2

Lγµψ2Lγ

µψ1L − ψ2

Rγµψ2Rγ

µψ1L

)= 0 (101)

iγµ∇µψ2L + 3

16

(ψ1Lγµψ

1Lγ

µψ2L + ψ2

Lγµψ2Lγ

µψ2L − ψ2

Rγµψ2Rγ

µψ2L

)= 0 (102)

iγµ∇µψ2R − 3

16

(ψ1Lγµψ

1Lγ

µψ2R + ψ2

Lγµψ2Lγ

µψ2R − ψ2

Rγµψ2Rγ

µψ2R

)= 0 (103)

which can be written in the form

iγµ∇µL− 12g~Aµ · ~σγµL+ 1

2g′BµγµL−GY φR = 0 (104)

iγµ∇µR+ g′BµγµR−GY φ†L = 0 (105)

by defining

(ψ2R

)= R

(ψ1L

ψ2L

)= L (106)

with

38 RL = φ (107)

and

38

(Lγµ

I

2L+ RγµR)= −

(g′

1−GY

)Bµ (108)

38 Lγµ

~σ2L =

(3g

3−GY

)~Aµ (109)

19

as new fields. We see that field equations (104) and (105) are formally identical to the field equations for the leptonfields before the symmetry breaking in the Standard Model.We notice that from the fundamental lepton fields, it is possible to build up a complex singlet and a complex

doublet: from these, we also see that the composite scalar field is a complex doublet; and the composite vector fieldsare a real singlet and a real triplet. In fact, consider that the massless fundamental leptons R and L are complex andtherefore they can be subject to two independent U(1) transformations: the massless composite scalar field, given bythe definition 3

8 RL, is consequently complex and thus it transforms according to the combined U(1) transformations.Finally the massless composite vector fields are real and do not transform at all. Moreover we have that the masslessfundamental leptons R and L are such that R is the only right-handed spinor and it does not transform into anythingelse whereas L is formed by a doublet of left-handed spinors which, in principle, are indistinguishable and thereforethey can transform into one another according to an SU(2)L transformation: the massless composite scalar field 3

8 RLtransforms according to the SU(2)L transformation above; furthermore, the massless composite vector fields are suchthat the vector field, given by the right-handed projections RγµR, does not transform whereas, on the other hand,the vector fields, given by the left-handed projections, are such that the vector field, given by LγµL, transforms intoitself. The three vector fields Lγµ~σL transform into each other and so the vector fields given by 2RγµR+ LγµL andLγµ~σL transform as the singlet and the triplet of the adjoint representation of the same SU(2)L transformation thathas been given above. This establishes the transformation laws of the lepton, the scalar and vector fields before thesymmetry breaking in the Standard Model.

C. The massless spinor-spinor system

As a second example, we shall deal with the case in which two spinors have both projections, but for which the tworight-handed projections have different Abelian charges so to be forbidden to mix [97]. The aim is to compare thiscase with that of the electroweak interactions for quarks before the symmetry breaking.In the case of massless spinors having both projections, there is nothing that would forbid the two right-hand

projections to mix forming a doublet, contrary to what is observed; to avoid this situation maintaining the two right-handed projections as two singlets, we have to postulate in addition that Abelian fields are introduced and the chargesare assigned to the two right-handed fields with two different values. The field equations then are

iγµ∇µψ1L + 3

16

(ψ1Lγµψ

1Lγ

µψ1L + ψ2

Lγµψ2Lγ

µψ1L − ψ1

Rγµψ1Rγ

µψ1L − ψ2

Rγµψ2Rγ

µψ1L

)= 0 (110)

iγµ∇µψ1R − 3

16

(ψ1Lγµψ

1Lγ

µψ1R + ψ2

Lγµψ2Lγ

µψ1R − ψ1

Rγµψ1Rγ

µψ1R − ψ2

Rγµψ2Rγ

µψ1R

)= 0 (111)

iγµ∇µψ2L + 3

16

(ψ1Lγµψ

1Lγ

µψ2L + ψ2

Lγµψ2Lγ

µψ2L − ψ1

Rγµψ1Rγ

µψ2L − ψ2

Rγµψ2Rγ

µψ2L

)= 0 (112)

iγµ∇µψ2R − 3

16

(ψ1Lγµψ

1Lγ

µψ2R + ψ2

Lγµψ2Lγ

µψ2R − ψ1

Rγµψ1Rγ

µψ2R − ψ2

Rγµψ2Rγ

µψ2R

)= 0 (113)

which may be written in the form

iγµ∇µL− 12g~σ · ~AµγµL− 1

6g′BµγµL−Guiσ

2φ∗u−Gdφd+O3 = 0 (114)

iγµ∇µu− 23g

′Bµγµu+GuiφTσ2L+O3 = 0 (115)

iγµ∇µd+13g

′Bµγµd−Gdφ†L+O3 = 0 (116)

when we define

(ψ1R

)= u

(ψ2R

)= d

(ψ1L

ψ2L

)= L (117)

with

1532Gd

(dL

)+ 3i

16Gu

(uσ2L

)∗= φ (118)

and

932

(Lγµ

I

2L+ 2uγµu− dγµd)+ qΓµ = g′Bµ (119)

932 Lγµ

~σ2L = g ~Aµ (120)

as new fields. In the field equations (114), (115) and (116), the terms O3 represent residual interactions written in theform of three-field vertices. Under the reasonable assumption for which these terms may be negligible, this is formallyidentical to the system of field equations for the quark fields before the symmetry breaking in the Standard Model.

20

The Abelian field Γµ must be fundamental and its coupling is given in terms of the − 16q, − 2

3q and 13q charges for

the left-handed L, right-handed u and d quarks respectively; the Abelian field, given by the current 932 (Lγµ

I

2L +

2uγµu − dγµd), is composite in terms of those quarks and what allows the present prescription to work is that thiscurrent is defined once but its appearance occurs in terms of the − 1

6 , − 23 and 1

3 factors in front of the L, u and d

quarks respectively: this is the reason for which the current 932 (Lγµ

I

2L + 2uγµu − dγµd) and the Abelian field qΓµmay always be reabsorbed into the definition of the Abelian field g′Bµ still fundamental and still occurring with the− 1

6 , − 23 and 1

3 factors in front of the L, u and d quarks again respectively. By following the same method presentedabove, it is possible to establish the form of the transformation laws for the quarks, the scalar and vector fields beforethe symmetry breaking in the Standard Model.

D. The massive spinor-semispinor system

In this final example, we shall deal with the case in which one of the two spinors is massive while the other ismassless and taken to be a semispinor having only the left-handed projection [98]. The aim is to compare this casewith that of the electroweak interactions for leptons in a situation in which there is no symmetry breaking at all.In the case is which one is a massive spinor and the other a massless semispinor having only the left-handed

projection, the two different masses prevent them to mix forming doublets; this situation is such that the two fieldsare always asymmetric and they have to be identified with the electron and neutrino field right from the beginning.In this case the field equations for the electron and neutrino fields e and ν are

iγµ∇µe− 316 (eγµeγ

µe+ νγµνγµγ5e)−me = 0 (121)

iγµ∇µν − 316eγµγ5eγ

µν = 0 (122)

which can be transformed into the following

iγµ∇µe− 38 (cos θ)

2eγµeγµe+ q tan θZµγµe− g

2 cos θZµγµeL + g√

2W ∗µγ

µν −me = 0 (123)

iγµ∇µν +g

2 cos θZµγµν + g√

2Wµγ

µeL = 0 (124)

once we define

Zµ = −[2(sin θ)2eγµe− eLγ

µeL + νγµν] (

3 cot θ16q

)(125)

Wµ = − (eLγµν)

[12(sin θ)2−3

16q√2 sin θ

](126)

as the new fields.The field equations (123) and (124) are formally identical to the system of field equations for the lepton fields after

the symmetry breaking and mass generation in the Standard Model; however, there has been no generation of anymass nor breaking of any symmetry because there has never been any symmetry in the model while there has alwaysbeen mass for the fermion fields.The lack of symmetry of the mediators comes from the fact that they are composite states of massive spinors due

to the presence of torsion, undergoing to no transformation law. The lack of the standard Higgs field comes from thefact that, in this approach, it would be useless. These two features constitute important discrepancies with respectto the Standard Model: in fact, in this example, we must expect the weak bosons to display internal structure whenthe energy is high enough to probe their compositeness while in the Standard Model they must remain structurelessat any energy scale; and because the present model predicts the absence of the Higgs field while the Standard Modelneeds its presence, then the detection of the Higgs or its persistent elusive character would settle the controversydefinitively.As an additional comment, we have to specify that, in this construction, there is a fundamental difference between

the case of leptons and that of quarks, that is in the former, the weak interaction are reproduced exactly, whereas, inthe latter, they have been reproduced only partially becoming exact when the three-field interactions are neglected:if, on the one hand, the corrections we have in the case of quarks may well be negligible, especially in situations inwhich also chromodynamics is present and where all quarks are confined as partons within nucleons, on the other handthere must be no correction for leptons since their interactions are relatively simple. In other words, we may say thatin a comparison between the present approach and the Standard Model, we may have small deviations in situationsin which the dynamics is dirty enough to keep them hidden, such as for quarks, but no deviation in situation thedynamics is clean enough to display them clearly, such as for leptons. Thus said, it is obvious how for our approachto work, the neutrino must be left-handed solely, and this means in particular massless. This situation may appear

21

to be in contrast with what is becoming the accepted paradigm for which neutrinos appear instead to be massive;however the mass of neutrinos is seen as a consequence of the theory that links mass to oscillation and the observationof the oscillations themselves, the last being an irrefutable experimental evidence while the first being a theory thatstill has with many conceptual problems: these two problems may be solved at once by a model in which neutrinoare massless and nevertheless still capable of oscillations. Incidentally, we do not have to look much far to find such amodel, because in presence of torsion, neutrinos do oscillate precisely because they are massless, as discussed in [99].Finally, we would like to spend a few words about the differences between the case in which fermions are initially

massless yielding the dynamics of the Standard Model before the symmetry breaking, and that in which they arealready massive yielding the dynamics of the Standard Model after the symmetry breaking; what we would like tospecify is that, in this last instance, the field equations are those that correspond to that of the Standard Model afterthe symmetry breaking, not because we have witnessed the breakdown of an initial symmetry but rather becausethe system has never been symmetric. In this last example the problem of having a symmetry breaking will becircumvented.These results show that torsion induces interactions among spinors whose form recalls that of the weak forces

between fermions: what this implies is that for the Dirac fermionic field equations, the presence of torsion would bemanifested disguised by weak forces: therefore the information about the weak forces is stored within torsion, in asimilar way as the information about gravitation is stored within the metric; this suggests that the weak forces andgravity (although intrinsically different) may be accommodated into (different) parts of the same connection. This isnot what we usually call unification, and yet it clearly is a stance of unification, inasmuch as a conceptual perspectiveis adopted.

IX. CONCLUSIONS

We have shown that, starting from a theory of gravity formulated in 5-dimensions, the 4D-spacetime reductionprocedure gives rise to extensions of GR in which the additional gravitational degrees of freedom can be consideredas additional physical scalar fields. Such scalar fields are geometrical deformations undergoing the GL(4)-group ofdiffeomorphisms and are capable of generating the masses of particles. Furthermore, the f(R) non-linearities encodeenergy-dependent running couplings able to fine-tune the scale. Finally, torsion gives rise to forces having the structureof weak force. In other words, for Dirac fermionic field equations, the presence of torsion would be manifested in theguise of weak forces: this means that, contrary to what is commonly believed, torsion might not be a quantity wenever observed due to its weakness, instead, it might actually be a quantity whose effects are constantly observed inweak experiments, although we were not ready to think weak forces as a long-range manifestation of torsion. If thiswere true, it would give rise to some sort of unification of weak and gravitational forces, as they may be includedrespectively in torsion and metric degrees of freedom, that is in different parts of the same connection. This connectionis naturally accompanied by electrodynamics, within the most general spinorial connection of the spinor field. Theonly thing that would be left out of this approach would be chromodynamics. However, this does not constitute abad point of the present scheme compared to the commonly accepted Standard Model: in fact, neither here nor inthe Standard Model chromodynamics is unified with all other interactions, and actually in the Standard Model, theweak and electrodynamic interactions are not unified as well, since the U(1)× SU(2) group, and for that matter thefull U(1) × SU(2) × SU(3) group are no unification of sub-groups into a larger group, but only a mere albeit quiteelegant patching of sub-groups.The philosophical implications of this approach are profound, as this would force us to rethink the issue of unification

in physics under a conceptual perspective, rather than according to the usual mathematical symmetry-based way,a way that, despite the plethora of possible symmetry groups, we have been investigating in the last forty years,has given no real advance since the SU(5)-model predicted the decay of the proton, yet to be discovered. In thisprescription there is no need to postulate any new ingredient, because torsion is naturally part of the most generalconnection, the non-linearity of the f(R) function comes out from the requirement that the action of gravitationalfield be not restricted to be the simplest Hilbert-Einstein action. If the action has to be written in terms of theRicci scalar curvature in the most general way, then f(R) where R contains torsion does constitute the most generaldynamics we may have for the gravitational background.However, some crucial points have to be considered. On the one hand, torsion manufactures weak interactions,

which, for fermions, are structurally identical to those arising from the Standard Model, and therefore any effect inthe dynamics of fermions would be reproduced as in the Standard Model itself. Nevertheless, the gauge fields emergeas composite fields and, although at energies that are low enough not to probe their internal compositeness, theyshould look like those of the Standard Model. However, for higher energies discrepancies, the Standard Model mustpop out. It is worth stressing that, in our approach, there is no standard Higgs field but a sort of gravitational Higgsmechanism is present. The validity of the presented scheme could be reasonably checked at LHC in short time, due

22

to the increasing luminosities of the set-up. At the present stage, experiments are indicating, very preliminary, thepresence of resonances and condensate states that cannot be considered final evidences for Higgs boson (at Fermi Lab,by CDF collaboration, and at CERN, in LHC, by CMS and ATLAS [92]).Incoming years will offer new data carrying information about the structure of the elementary constituents of our

universe, and although the full confirmation of the Standard Model and its most accredited extensions are expected,nevertheless they may fail to come; if so, new aspects of the problem of unification in physics will have to be faced,and they may turn out to be even more exciting. And despite the direction taken since the 1960s, the dream ofunification in physics might be much more similar to what Einstein himself hoped it to be.

[1] G. Basini and S. Capozziello, ”Open Quantum Relativity”, in Classical and Quantum Gravity Research (Ed. Christiansen,Nova Science Publishers, New York, 2008).

[2] G. Basini and S. Capozziello, Gen. Rel. Grav., 35, 2217 (2003).[3] C. Rovelli, Quantum Gravity, Cambridge University Press, Cambridge (2004).[4] G. Basini and S. Capozziello, Mod. Phys. Lett. A 20, 251 (2005).[5] G. Basini and S. Capozziello, Int. Jou. Mod. Phys. D 15, 583 (2006).[6] R. Utiyama, Phys. Rev. 101, 1597 (1956).[7] C. N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954).[8] T.W. Kibble, J. Math. Phys. 2, 212 (1960).[9] E. Cartan, Ann. Ec. Norm. 42, 17 (1925).

[10] D.W. Sciama, On the analog between charge and spin in GR, in Recent Developments in GR, Festschrift for Leopold Infeld,(1962) 415, Pergamon Press, New York.

[11] R. Finkelstein, Ann. Phys. 12, 200 (1961).[12] F. W. Hehl et al., J. Math. Phys. 12,1334 (1970).[13] F.W. Hehl et al., Rev. Mod. Phys. 48, 393 (1976).[14] F. Mansouri et. al., Phys. Rev. D13, 3192 (1976) .[15] F. Mansouri, Phys. Rev. Lett. 42, 1021 (1979).[16] L. N. Chang et al., Phys. Rev. D13, 235 (1976).[17] G. Grignani et. al., Phys. Rev. D45, 2719 (1992).[18] F. W. Hehl and J. D. McCrea, Found. Phys. 16, 267 (1986).[19] H. I. Arcos and J. G. Pereira, Int. J. Mod.Phys. D 13, 2193 (2004).[20] A. Inomata et. al., Phys. Rev. D19 , 1665 (1978).[21] C. G. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 117, 2247 (1969).[22] S. Coleman, J. Wess and B. Zumino, Phys. Rev. 117, 2239 (1969).[23] C. J. Isham, A. Salam and J. Strathdee, Ann. of Phys. 62, 98 (1971).[24] A.B. Borisov and V.I. Ogievetskii, Theor. Mat. Fiz. 21, 329 (1974).[25] E. A. Ivanov and V. I. Ogievetskii, Gauge theories as theories of spontaneous breakdown, Preprint of the Joint Institute of

Nuclear Research, E2-9822 3-10 (1976).[26] L. N. Chang et al., Phys. Rev. D17, 3168 (1978).[27] K. S. Stelle et al., Phys. Rev. D21, 1466 (1980).[28] E. A. Ivanov and J. Niederle, Phys. Rev. D 25, 976 (1982).[29] E. A. Ivanov and J. Niederle, Phys. Rev. D 25, 988 (1982).[30] D. Ivanenko and G. A. Sardanashvily, Phys. Rep. 94 , 1 (1983).[31] A. Lord and P. Goswami, J. Math. Phys. 27, 3051 (1986).[32] E. A. Lord and P. Goswami, J. Math. Phys. 29, 258 (1987).[33] Y. Ne’eman and T. Regge, Riv. Nuovo Cimento 1 1, (1978).[34] A. Lopez-Pinto, A. Tiemblo, R. Tresguerres, Class. Quant. Grav. 12, 1503 (1995).[35] J. Julve et. al., Class. Quantum Grav. 12, 1327 (1995).[36] R. Tresguerres and E. W. Mielke, Phys.Rev. D62, 044004 (2000).[37] R. Tresguerres, Phys. Rev. D 66, 064025 (2002).[38] A. Tiemblo and R. Tresguerres, Eur.Phys.J. C42, 437 (2005).[39] A. Tiemblo and R. Tresguerres, Recent Res.Devel.Phys. 5, 1255 (2004).[40] T. Appelquist, A. Chodos, P.G.O. Freund, Modern Kaluza-Klein Thories, Frontiers in Physics, Addison-Wesley Publishing

Company, Inc. Redwood, CA (1987).[41] G.G. Ross, Grand Unified Theories, Benjamin, Menlo Park, CA (1985).[42] G. Basini, A. Morselli, M. Ricci, La Riv. del N. Cimento 12 (1989).[43] S. Capozziello, G. Lambiase, C. Stornaiolo, Ann. Phys. (Leipzig) 10, 713 (2001).[44] E. Kaehler, Rendiconti di Matematica 21, 425 (1962).[45] V. De Sabbata and C. Sivaram, “Spin and Torsion in Gravitation”

(World Scientific, Singapore, 1994).[46] Y. Nambu and G. Jona–Lasinio, Phys. Rev. 124, 246 (1961).

23

[47] L. Fabbri and S. Vignolo, Class. Quantum Grav. 28, 125002 (2011).[48] S. Capozziello, G. Basini, M. De Laurentis, Eur. Phys. J. C 71, 1679 (2011).[49] A. Skopenkov, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes.

347, London (2008).[50] S. Capozziello and V. Faraoni, Beyond Einstein Gravity, Fundamental Theories of Physics Vol. 170, Springer Ed., Dordrecht

(2011).[51] S. Capozziello, M. De Laurentis, Phys. Repts. 509, 167 (2011).[52] S. Nojiri, S.D. Odintsov, Phys. Reps. 505, 59 (2011).[53] B. Coll, J. Llosa and D. Soler, Gen. Rel. Grav. 34, 269 (2002).[54] S. Capozziello and C. Stornaiolo, Int. Jou. Geom. Meth. Mod. Phys. 5, 185 (2008).[55] R. M. Wald, GR, The University of Chicago Press, (1984).[56] J.E. Campbell A Course of Differential Geometry, Clarendon, Oxford (1926).[57] S. Capozziello, J. Matsumoto, S. Nojiri, S. D. Odintsov, Phys. Lett. B 693, 198 (2010).[58] N.Birrell and P.C.Davies, Quantum Fields in Curved Space, Cambridge Univ. Cambridge (1984).[59] A. Pich, arXiv: 0705.4264[hep-ph] (2007).[60] I. Antoniadis, S. Dimopoulos and A. Giveon, JHEP 0105, 055 (2001)[61] X. Calmet, S. D. H. Hsu, D. Reeb Phys. Rev. D 77, 125015 (2008).[62] P. Minkowski, Phys. Lett. B 71, 419 (1977).[63] A. Zee, Phys. Rev. Lett. 42, 417 (1979).[64] X. Calmet, S. D. H. Hsu, Phys. Lett. B 663, 95 (2008).[65] X. Calmet, Mod. Phys. Lett. A 26, 1571 (2011).[66] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 436, 257 (1998).[67] S. Capozziello, R. Cianci, C. Stornaiolo, S. Vignolo, Class. Quant. Grav. 24, 6417 (2007).[68] P. Meade and L. Randall, Jou. High. Energy Phys. 0805, 003, 2008)[69] J. L. Feng and A. D. Shapere, Phys. Rev. Lett. 88, 021303 (2002)[70] L. A. Anchordoqui, J. L. Feng, H. Goldberg and A. D. Shapere, Phys. Rev. D 65, 124027 (2002)[71] L. A. Anchordoqui, J. L. Feng, H. Goldberg and A. D. Shapere, Phys. Rev. D 68, 104025 (2003)[72] A. Ringwald and H. Tu, Phys. Lett. B 525, 135 (2002).[73] M. Kowalski, A. Ringwald and H. Tu, Phys. Lett. B 529, 1 (2002)[74] C. H. Llewellyn Smith, Phys. Lett. B 46, 233 (1973) .[75] B. W. Lee, C. Quigg and H. B. Thacker, Phys. Rev. Lett. 38, 883 (1977).[76] B. W. Lee, C. Quigg and H. B. Thacker, Phys. Rev. D 16, 1519 (1977).[77] C. E. Vayonakis, Lett. Nuovo Cim. 17, 383 (1976).[78] G. Dvali, G. F. Giudice, C. Gomez and A. Kehagias, CERN-PH-TH-226 (2010).[79] V. Visnjic, Nuovo Cim. A 101, 385 (1989).[80] G. ’t Hooft, “Topological aspects of quantum chromodynamics”, Lectures given at International School of Nuclear Physics,

Erice, Italy, 17-25 Sep. 1998; published in “Erice 1998, From the Planck length to the Hubble radius”.[81] G. ’t Hooft, in “Recent Developments In Gauge Theories,” Cargese 1979, ed. G. ’t Hooft et al. Plenum Press, New York,

1980, Lecture II, p.117; proceedings, Nato Advanced Study Institute, Cargese, France, August 26 - September 8, 1979.”[82] S. Weinberg, in Understanding the Fundamental Constituents of Matter, ed. A. Zichichi (Plenum Press, New York, 1977).[83] J. Hewett and T. Rizzo, JHEP 0712, 009 (2007).[84] H. Arason, D. J. Castano, B. Keszthelyi, S. Mikaelian, E. J. Piard, P. Ramond and B. D. Wright, Phys. Rev. D 46, 3945

(1992).[85] M. Fabbrichesi, R. Percacci, A. Tonero, O. Zanusso, Phys.Rev. D83, 025016 (2011).[86] R. Percacci and O. Zanusso, Phys. Rev. D 81, 065012 (2010)[87] T. Appelquist and M. S. Chanowitz, Phys. Rev. Lett. 59, 2405 (1987).[88] F. Maltoni, J. M. Niczyporuk and S. Willenbrock, Phys. Rev. Lett. 86, 212 (2001).[89] F. Maltoni, J. M. Niczyporuk and S. Willenbrock, Phys. Rev. D 65, 033004 (2002).[90] D. A. Dicus and H. J. He, Phys. Rev. D 71, 093009 (2005).[91] D. A. Dicus and H. J. He, Phys. Rev. Lett. 94, 221802 (2005).[92] G. Aad et al. (The ATLAS collaboration) Eur. Phys. J. C 71, 1512 (2011).[93] V. De Sabbata and M. Gasperini, Gen. Rel. Grav. 10, 731 (1979).[94] C. Sivaram and K. P. Sinha, Lett. Nuovo Cim. 13, 357 (1975).[95] C. Sivaram and K. P. Sinha, Phys. Rept. 51, 111 (1979).[96] L. Fabbri, Mod. Phys. Lett. A 26, 1697 (2011).[97] L. Fabbri, arXiv:1009.4423 [hep-th].[98] L. Fabbri, Int. J. Theor. Phys. 50, 3616 (2011).[99] L. Fabbri, arXiv:1012.3862 [hep-th].